Unit 05 · Lesson 07 · Week 15 · Differentiation Rules & Higher Derivatives
Derivatives of Logarithmic Functions
Logarithmic derivatives convert multiplicative structure into reciprocal rates. Domain restrictions remain essential because real logarithms require positive arguments. This lesson emphasizes derivatives of logarithmic functions through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in derivatives of logarithmic functions.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do derivative rules make complex rates of change efficient, interpretable, and verifiable?
Visual model
Logarithmic derivatives convert multiplicative structure into reciprocal rates. Domain restrictions remain essential because real logarithms require positive arguments.
(\ln x)'=\frac1x(\log_a x)'=\frac1{x\ln a}
Worked reasoning
Misconception check: The derivative of ln(g(x)) is not 1/g(x); the inner derivative is required.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Derivatives of Logarithmic Functions
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Differentiate ln x.
Answer: 1/x
Reasoning: The real-domain restriction is x>0.
Reasoning: The real-domain restriction is x>0.
Differentiate ln(x²+1).
Answer: 2x/(x²+1)
Reasoning: Use the chain rule.
Reasoning: Use the chain rule.
Differentiate log₁₀x.
Answer: 1/(x ln10)
Reasoning: A base-10 logarithm includes ln10.
Reasoning: A base-10 logarithm includes ln10.
Find the slope of y=ln x at x=1.
Answer: 1
Reasoning: 1/x evaluated at 1 is 1.
Reasoning: 1/x evaluated at 1 is 1.
Differentiate x ln x.
Answer: ln x+1
Reasoning: Use the product rule.
Reasoning: Use the product rule.
Why must the log argument be checked?
Answer: Real logarithms require a positive argument.
Reasoning: The derivative formula does not remove the original domain restriction.
Reasoning: The derivative formula does not remove the original domain restriction.